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Theorem fsum0diaglem 12255
Description: Lemma for fsum0diag 12256. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
fsum0diaglem  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( k  e.  ( 0 ... N
)  /\  j  e.  ( 0 ... ( N  -  k )
) ) )
Distinct variable group:    j, k, N

Proof of Theorem fsum0diaglem
StepHypRef Expression
1 elfzle1 10815 . . . . . . 7  |-  ( j  e.  ( 0 ... N )  ->  0  <_  j )
21adantr 451 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  0  <_  j
)
3 elfz3nn0 10839 . . . . . . . . . 10  |-  ( j  e.  ( 0 ... N )  ->  N  e.  NN0 )
43adantr 451 . . . . . . . . 9  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  NN0 )
54nn0zd 10131 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  ZZ )
65zred 10133 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  RR )
7 elfzelz 10814 . . . . . . . . 9  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ZZ )
87adantr 451 . . . . . . . 8  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ZZ )
98zred 10133 . . . . . . 7  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  RR )
106, 9subge02d 9380 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0  <_ 
j  <->  ( N  -  j )  <_  N
) )
112, 10mpbid 201 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  <_  N
)
125, 8zsubcld 10138 . . . . . 6  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  j )  e.  ZZ )
13 eluz 10257 . . . . . 6  |-  ( ( ( N  -  j
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  ( N  -  j ) )  <->  ( N  -  j )  <_  N ) )
1412, 5, 13syl2anc 642 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  e.  ( ZZ>= `  ( N  -  j ) )  <-> 
( N  -  j
)  <_  N )
)
1511, 14mpbird 223 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  N  e.  (
ZZ>= `  ( N  -  j ) ) )
16 fzss2 10847 . . . 4  |-  ( N  e.  ( ZZ>= `  ( N  -  j )
)  ->  ( 0 ... ( N  -  j ) )  C_  ( 0 ... N
) )
1715, 16syl 15 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( 0 ... ( N  -  j
) )  C_  (
0 ... N ) )
18 simpr 447 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... ( N  -  j ) ) )
1917, 18sseldd 3194 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ( 0 ... N ) )
20 elfzelz 10814 . . . . . 6  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  e.  ZZ )
2120adantl 452 . . . . 5  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  ZZ )
2221zred 10133 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  e.  RR )
23 elfzle2 10816 . . . . 5  |-  ( k  e.  ( 0 ... ( N  -  j
) )  ->  k  <_  ( N  -  j
) )
2423adantl 452 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  k  <_  ( N  -  j )
)
2522, 6, 9, 24lesubd 9392 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  <_  ( N  -  k )
)
26 elfzuz 10810 . . . . 5  |-  ( j  e.  ( 0 ... N )  ->  j  e.  ( ZZ>= `  0 )
)
2726adantr 451 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  (
ZZ>= `  0 ) )
285, 21zsubcld 10138 . . . 4  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( N  -  k )  e.  ZZ )
29 elfz5 10806 . . . 4  |-  ( ( j  e.  ( ZZ>= ` 
0 )  /\  ( N  -  k )  e.  ZZ )  ->  (
j  e.  ( 0 ... ( N  -  k ) )  <->  j  <_  ( N  -  k ) ) )
3027, 28, 29syl2anc 642 . . 3  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( j  e.  ( 0 ... ( N  -  k )
)  <->  j  <_  ( N  -  k )
) )
3125, 30mpbird 223 . 2  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  j  e.  ( 0 ... ( N  -  k ) ) )
3219, 31jca 518 1  |-  ( ( j  e.  ( 0 ... N )  /\  k  e.  ( 0 ... ( N  -  j ) ) )  ->  ( k  e.  ( 0 ... N
)  /\  j  e.  ( 0 ... ( N  -  k )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0cc0 8753    <_ cle 8884    - cmin 9053   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798
This theorem is referenced by:  fsum0diag  12256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799
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